Talk:Mathematics

Definition source

I am trying to find the origin or the author of

"Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself"

...but my search was fruitless so far. Any ideas? 217.77.54.213 (talk) 17:06, 12 October 2024 (UTC)

Mathematics Article Problems.

I am not yet an auto-confirmed user( Hellow Hellow i am here 19:01, 23 October 2024 (UTC) )but I spotted some problems. It shows the types of numbers, but is missing real and complex numbers, as well as imaginary numbers.

The article does discuss each of these topics. Remsense ‥ 论 19:07, 23 October 2024 (UTC)

Oh, sorry. I must have missed them. Hellow Hellow i am here 14:00, 25 October 2024 (UTC)

Number theory topline definition

@D.Lazard: Daniel: 90% of number theory is about the properties of algebraic numbers, and saying "numbers" in general is very misleading, since most study of real numbers occurs in analysis. "Whole numbers and fractions" are the main interest of number theory, and algebraic numbers appear as generalizations of them.

I wouldn't use "algebraic number" in the lead, but do you really think "whole number" and "fraction" are too technical, that they will confuse readers who are curious about mathematics, but do not know what whole numbers are?

Magyar25 (talk) 21:50, 5 November 2024 (UTC)

My take: If we need to gloss the term "number theory" at all, I would prefer "the theory of the natural numbers", which is accurate in spite of the fact that arithmeticians consider other sorts of numbers. Rationals are ratios of natural numbers. Algebraic numbers are algebraic over the natural numbers. Et cetera.

As to the term whole number, my preference would be that we should never use (as opposed to mention) it at all, especially in a math article. Mathematicians essentially never use the term. --Trovatore (talk) 22:18, 5 November 2024 (UTC)

The lead of the article Mathematics is not the place for an accurate definition of number theory. This sentence is here to explain what mathematics is about, and the way readers understand "number" does not really matter. Moreover, restricting number theory to some sort of numbers would go against a common consensus: the facts "every real number has an infinite decimal expansion" and "π is not an algebraic number" are properties of real numbers that everyone considers as belonging to number theory. If there is something that is misleading in this sentence it is the definition of analysis as "the study of continuous changes", since the phrase in rarely used in analysis, except for explaining one of several motivations of analysis. Nevertheless, after many discussions on this talk page, nobody has found a better phrase. D.Lazard (talk) 01:49, 6 November 2024 (UTC)

Sorry, I completely disagree with the claim the facts "every real number has an infinite decimal expansion" and "π is not an algebraic number" are properties of real numbers that everyone considers as belonging to number theory. I think that's just absolutely wrong. The first one is definitely not part of number theory. The one about π is a little closer but I still think it's unlikely to be called number theory. --Trovatore (talk) 19:01, 6 November 2024 (UTC)

Transcendental number theory and diophantine approximation are both part of number theory, fwiw. Tito Omburo (talk) 19:26, 6 November 2024 (UTC)

But π is not. --Trovatore (talk) 19:31, 6 November 2024 (UTC)

I mean, of course you use π in number theory, to give approximations and so forth. But you don't really study π. --Trovatore (talk) 19:32, 6 November 2024 (UTC)

The Borweins would disagree. Tito Omburo (talk) 19:34, 6 November 2024 (UTC)

Ref? --Trovatore (talk) 19:42, 6 November 2024 (UTC)

Pi and the AGM: a study in analytic number theory and computational complexity, Jonathan and Peter Borwein, 1987. Tito Omburo (talk) 19:45, 6 November 2024 (UTC)

Well. Categorizing branches is always fraught. I did say π was "a little closer". My general take is that nothing that involves the completed infinite is part of the subject matter of number theory, though it might be part of the methods.

Anyway the best solution might be just not to gloss "number theory" at all in the lead. I don't see that a gloss saying it's the "theory of numbers" adds anything at all; it just sounds like the natural meaning of the words. If we are to have a gloss I continue to think the "theory of natural numbers" is better wording. --Trovatore (talk) 20:12, 6 November 2024 (UTC)

p-adic numbers and adeles are unambiguously a part of number theory, and certainly involve completion. Tito Omburo (talk) 20:39, 6 November 2024 (UTC)

OK. I was never a number theorist, and maybe the field has moved on since I took my one class in it as an undergrad (we used Apostol's Introduction to Analytic Number Theory). I still don't find the current gloss useful. (Note that p-adic numbers and adele rings are not likely to be evoked by the phrase "the study of numbers".) Do you agree with just removing the gloss? --Trovatore (talk) 21:29, 6 November 2024 (UTC)

(edit conflict)The gloss for number theory is here for the balance of the sentence. If you can propose a better gloss, please do.

The first sentence of Transcendental number theory is "Transcendental number theory is a branch of number theory that investigates transcendental numbers". If you read the article, you will learn that a major result of this branch of number theory is Gelfond–Schneider theorem, which implies that ⁠$e^{\pi }$⁠ is trancendental, and that a major open question is whether ⁠$e+\pi$⁠ is transcendental.

About "completed infinity" in number theory: I never saw anybody writing that Fermat's Last Theorem and Wiles's proof of Fermat's Last Theorem do not belong to number theory, although the proof makes a fundamental implicit use of the axiom of infinity, and even (in the original proof) of a much stronger axiom. So, your opinion on the subject matter of number theory goes against a consensus of number theorists. D.Lazard (talk) 21:35, 6 November 2024 (UTC)

On the other hand, the first sentence of number theory says that it's about natural numbers and arithmetic functions. So there's a bit of a conflict there. Myself, I would not have counted transcendental number theory as part of number theory, but I don't know how workers in the field think about it.

My proposal is simply to have no gloss at all.

As to your second paragraph, you're talking about the proofs, not the subject matter. Fermat's last theorem is about natural numbers. Its proof uses completed infinite objects, but that is not what it is about. --Trovatore (talk) 22:30, 6 November 2024 (UTC)

I see that this conversation is moribund but I would definitely second this -- characterizing number theory as the "study of numbers" is extremely misleading. It would be preferable to list it without the parenthetical if we don't feel that it can be easily summarized.

Lauciusa (talk) 17:58, 25 March 2025 (UTC)

Small change suggestion

Change the begining of the article to be: Mathematics is a branch of knowledge and a field of study... Linking knowledge to not break the philosphy game Moondarkside01 (talk) 16:46, 6 November 2024 (UTC)

Thanks for your suggestion, but maintaining the philosophy game is not one of our goals. (If it were, then we would maintain the philosophy game, and it would be unsurprising that the philosophy game held, and then there would be no point to the philosophy game). Mgnbar (talk) 18:32, 6 November 2024 (UTC)

Semi-protected edit request on 15 December 2024

Hello, Wikipedia editors. I would like to request access to edit this article because of my interest in mathematics and my desire to contribute constructively to its content. I am passionate about the subject and eager to help improve the quality of this page, ensuring it remains accurate and informative. Slavuska Shabliy (talk) 17:42, 15 December 2024 (UTC)

Pages get "semi-protected" when they see repeated vandalism, edit warring, or other disruption from IP editors and/or new accounts. On a semi-protected page, brand new accounts can only make requests on the talk page for someone else to implement. But if you wait a few days and make several edits (counted across all of Wikipedia) your account will become "confirmed" and you can then make edits to semi-protected pages, including this one. The goal of the "semi-protection" is to solve the steepest part of the maintenance burden while not excessively restricting people from editing.

In the mean time (i.e. as an "unconfirmed" editor), if you have a specific change that you want to make here, you can make a specific request and someone can make that change to the article. If you want to make a more extensive change you can work on your desired text somewhere else, for instance in your user namespace, at a page like User:Slavuska Shabliy/mathematics, then come back to this talk page when you are ready for someone to apply those.. –jacobolus (t) 18:51, 15 December 2024 (UTC)

High compliments

The first paragraph of the lede here is excellent. Extreme kudos to everyone who contributed. HiDrNick! 22:51, 22 December 2024 (UTC)

Jain mathematics

Two editors attempted to add a long papagraph on Jain mathematics. This is misplaced here and gives an WP:UNDUE weight to this part of the history of mathematics.

Indeed the {{main article}} template at the beginning of § History say that this section is a summary of History of mathematics. This article does not mention Jain mathematics, but has a section on Indian mathematics with a template {{main article|Indian mathematics}}. Indian mathematics has several sections including a section on Janism. If more must be added on Janism, this must be done first in these "main articles".

Important contributions of Indian mathematics are already mentioned at the end of § Ancient. I have no opinion whether other important contributions must be mentioned there, but, if this requires more than a few words to mention them, this must be discussed first in this talk page. D.Lazard (talk) 12:03, 14 January 2025 (UTC)

Add reference to Albert Einstein in Mathematics#Reality section

I found the issue by reading an excerpt of this article in the Philosophy of Mathematics article. It looks a bit inconsistent, as all other enlisted scientists are referenced properly. Sunlamb (talk) 12:55, 25 January 2025 (UTC)

I guess that what you call a bad reference is the fact the mention of Einstein is not wikilinked to the article Albert Einstein. It was intensional, since the link appears already in two other sections. However, this excerpt has been inseterd more recently in Philosophy of mathematics. Also, readers are not supposed to read sections in their occurring order. So, I'll add the link. D.Lazard (talk) 15:02, 25 January 2025 (UTC)

Thank you Sunlamb (talk) 21:34, 22 March 2025 (UTC)

Definition and Introduction Proposal

The current definition reads

Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

This is awful, not least because it is circular. An operational definition of mathematics is basically impossible because of the huge variety of the field (there's even a section of the page on this), and what I would propose is skirting the issue by not proposing a definition.

Mathematics is a large and varied field of study that has its origins in the analysis and manipulation of numbers and shapes.

Mathematics is a large and varied field of study that has its origins in the analysis and manipulation of quantities and shapes.

And then continuing from there, with some examples of fields of study.

Lauciusa (talk) 17:54, 25 March 2025 (UTC)

I recommend searching the archives for "first sentence". It's written how it is for rather good reasons. Remsense ‥ 论 17:59, 25 March 2025 (UTC)

As Remsense alluded to, this sentence is highly contentious. And the supposed circularity doesn't bother me at all.

That said, I think that Lauciusa's solution is pretty good, and better than what we have. Lauciusa, your case would be strengthened if you could cite reliable sources (possibly tweaking the text according to them). Mgnbar (talk) 18:07, 25 March 2025 (UTC)

Cards on the table, I wouldn't agree. The present opening sentence is ponderous, but at least each word is considerably precise and meaningful. I'm skeptical of manipulation, which seems highly impressionistic—is tax collection in ancient Babylon "manipulation of numbers"? Remsense ‥ 论 18:15, 25 March 2025 (UTC)

I don't feel that the use of "manipulation" here is less precise and meaningful than the "needs of ... mathematics itself". There is literally nothing in the universe of objects and concepts that cannot be fit into this definition because there is no base case for the recursion (this is pedantic, I know, but we're editing a wikipedia article on mathematics so I think pedantic is where we are).

Is tax collection in ancient Babylon mathematics? I mean, I would argue that in terms of the use of numbers for record keeping and analysis, it is, yes. Certainly this plays into the origins of mathematics -- the early use of numbers and sums and products led to discoveries about those systems.

Lauciusa (talk) 14:16, 26 March 2025 (UTC)

That's going to be an argument one can make for every noun and verb we could use here. Ultimately, I'm pretty convinced we cannot write a more elegant, "non-circular" definition of what mathematics is because it simply would not reflect reality. Others may agree with me here—as human endeavors go (this characterization applies to some extent according to pretty much everyone, though some may quibble on margins for invention vs. discovery reasons?) mathematics is uniquely abstract and not always commensurable with "independent terms" (the connection to empirical science may be the best we can do), so the "feedback loop" you seem particularly bothered by, is to me a plain reflection of what mathematics has meant historically. I hope that makes sense. Remsense ‥ 论 14:35, 26 March 2025 (UTC)

In a larger sense, I am pretty convinced that we cannot write a concise definition of mathematics at all. Gun to my head, I would probably write a checklist of 20 features of mathematical study, and 20 anti-features of mathematical study, and if a field scored more than +9 then it would be considered within "mathematics".

But my proposal here is to precisely avoid getting pulled into a game of trying to create a definition at all. We give a general idea of the historical motivations for the study (which I think is unambiguously factual) and then in the next sentence we give some examples of modern mathematical subfields of study that have emerged from that.

I think where we disagree is that the existing definition is at all satisfactory. I feel that it is not. It is so broadly phrased (in the recursiveness) that it fails to exclude anything, but so restrictive in the remainder (specifically "the needs of empirical sciences") that it excludes the overwhelming majority of fields of mathematical endeavor.

There's a section of the article on the dispute around the definition and scope of mathematics; it is a field of study in itself (although more a field of semantics in my mind than anything prescriptive) and not worth trying to engage with in the first sentence.

Lauciusa (talk) 14:45, 26 March 2025 (UTC)

See, I agree we're not really defining mathematics in the traditional sense either way, but rather we are identifying it by its fruits. Yours is simply less helpful in doing so, though—large and varied field communicates very little at all, and manipulation of numbers and shapes is still offputtingly concrete in its vagueness.

Right now I do not see how any point in the present lead paragraph can actually be argued to be false, misleading, or even unhelpful to readers. It's not a good solution to answer the anxiety about lacking an airtight definition by resolving to say as little as possible instead. We can clearly say more of substance, and the present lead does. Remsense ‥ 论 14:55, 26 March 2025 (UTC)

I don't really disagree with your post, Remsense, but it exemplifies what I said earlier, about how editors disagree in their priorities. Your post is focused on factual content, rather than on making that content understandable to the reader.

In my opinion, the current opening sentence is unhelpful to readers because its structure is too complicated, especially for non-native speakers of English. The complication is what gives a lot of readers a knee-jerk "I don't like this" response. In general, we ask too much of the opening sentence. Mgnbar (talk) 15:02, 26 March 2025 (UTC)

I agree in principle also—its diction may well be worth tinkering with. As you say, if we're doing either that or a more drastic downsizing (which I'm not presently in favor of) we should be clear about what issues we're trying to solve. Remsense ‥ 论 15:07, 26 March 2025 (UTC)

This seems like a good avenue; I'll see what I can dig up. The "large and diverse field" I can probably cite much of the same material as in Definition of mathematics since that article directly addresses the controversy in defining the field.

For the historical origins of mathematics (i.e. numbers & shapes as I have proposed here) it shouldn't be hard to find supporting material in the introduction of any survey of mathematics. If you have suggestions I'm open to them, otherwise I'll dig in further when I can.

Lauciusa (talk) 15:26, 26 March 2025 (UTC)

The present first sentence is a compromise that could certainly be improved. Nevertheless, Lauciusa's proposal is much worse, as it hides the most fundamental features of mathematics: All areas of mathematics involve theories, theorems and proofs and the relatioship between mathematics and empirical sciences is fundamental. So, I am strongly against Lauciusa's proposal. D.Lazard (talk) 18:32, 25 March 2025 (UTC)

Lauciusa, these responses give you an idea of what you're up against. People have high standards for the opening sentence: verifiability, accuracy, clarity, completeness, etc. We have not found a sentence that achieves all of these goals, and we disagree about which goals to abandon first, etc. (I hope that I am not mis-representing any of my fellow editors here. I write this post with sincerity and empathy.) Regards, Mgnbar (talk) 19:13, 25 March 2025 (UTC)

I can live with both the current first sentence and with Lauciusa's proposal, and don't have a strong preference between them.

I will say that one clear advantage of Lauciusa's proposal is that it at least says something about the objects of study of mathematics, which the current first sentence does not.

As an aside, I don't think I completely agree with Prof Lazard's claim that "all" areas of mathematics involve theorems and proofs; that seems shading towards a Euclidean POV. It's true as a practical matter for research mathematics today, but there's lots of stuff that I would consider "mathematics" for which it's a problematic claim. --Trovatore (talk) 20:03, 25 March 2025 (UTC)

I think the lack of "study objects" in the first sentence is nicely made moot by the following two sentences—sometimes I feel I drill down too fractally to the level of the sentence when judging how material is weighed in the lead. Remsense ‥ 论 20:06, 25 March 2025 (UTC)

Can you give an example for something you'd consider "mathematics" which does not involve theorems? –jacobolus (t) 20:38, 25 March 2025 (UTC)

Well, its ancient origins, for example, before the notion of "theorem" was really formulated. Or similar explorations even today (for example in recreational mathematics or experimental mathematics). --Trovatore (talk) 20:42, 25 March 2025 (UTC)

Let me start by saying that my proposed change does not contradict any of this, but is simply a factual statement about the origin of mathematics as a field, and is non-controversial as far as I'm aware (although also admittedly not comprehensive by design)

The idea that all areas of mathematics involves theories, theorems, and proofs does not seem accurate to me. Calculation and arithmetic, while distasteful to the practicing mathematician, are clearly part of mathematics, and arguably are the foundation upon which all of mathematics is motivated.

In terms of the relationship of mathematics and empirical sciences, I would broadly say an even stronger converse -- mathematics is more accurately the study of things that cannot be contradicted by empirical methods!

Lauciusa (talk) 14:03, 26 March 2025 (UTC)

Most mathematics concerns the sciences, not "numbers and shapes". Try again. Tito Omburo (talk) 21:46, 25 March 2025 (UTC)

First, I want to say that my proposed opening does not contradict this. Even if we grant that mathematics concerns the sciences, its origins in the study of arithmetic and geometry are hard to dispute.

But as a direct response, I don't think this is true; mathematics in most cases distinguishes itself from the sciences based almost entirely on the fact that it is not an empirical examination; it cannot be contradicted by empirical study.

Lauciusa (talk) 14:07, 26 March 2025 (UTC)

This is exacly what is said in the current first sentence: mathematics is not an empirical science but most mathematics were developped as tools for empirical sciences and for mathematics itself. Also, it is arguable wether the empirical rules developed before Greek mathematics were mathematics. In any case, this is Greek mathematicians who coined the word "mathematics" and elaborated the concept of proof. D.Lazard (talk) 15:08, 26 March 2025 (UTC)

See, I think this is a bit POV, specifically a Euclidean-foundationalist POV. In my view, mathematics is distinguished by its subject matter (mathematical objects) more than by its methods (proof being the method currently under scrutiny, though not the only method used in mathematics). There used to be a first sentence that was more explicitly about the subject matter (in various versions it had a list of things ...including quantity, space, pattern, and change... or something like that). It was less elegant than the current first sentence, but it did have the advantage of bringing the subject matter front and center. --Trovatore (talk) 16:53, 26 March 2025 (UTC)

Of course, and at the same time it's nontrivial to really discern what the ideal, modern, internally consistent notion of a concept (here, μάθημα) should be when “you” were the ones with which the word historically emerged to refer to a particular historically contingent but certainly antecedent sense of what it means now.

(I'm currently doing a lot of research for Logic in China, which seems a bit more cut and dry as being logic unless we really think a tradition doesn't count unless it assigns special symbols while analyzing methods of reasoning and argumentation.) Remsense ‥ 论 17:56, 26 March 2025 (UTC)

As I've said before, the current opening is both overly expansive and overly restrictive in ways that only make sense if you already have an intuition of what mathematics is. Microscopes were developed as tools for empirical sciences. Why is optics not part of mathematics? Note that I am not saying that the definition implies the inverse, that all tools for empirical sciences are mathematics; just that it fails to add clarity on what exactly distinguishes mathematics from other tools.

I don't think we can provide a clear definition here, and I don't think we should try to create one. The definition provided is bad and inaccurate.

I do not propose an alternate definition, very deliberately. Instead we open with historical context and follow up with broad strokes on where it lies today.

If you want to say that arithmetic and the development of counting are not part of mathematics, then I most strenuously disagree, and I think any realistic look at primary and secondary mathematical education means that I am not alone in this disagreement.

Lauciusa (talk) 15:20, 26 March 2025 (UTC)

The first sentence is not a definition of mathematics. I'm unclear why your proposal is less of a definition. The rest of the first paragraph provides added context. Tito Omburo (talk) 15:34, 26 March 2025 (UTC)

I'm not sure I understand. The existing first sentence looks like definition to me. It defines what "mathematics" is. Is there something subtle I'm missing? It attempts to be an exhaustive description of what mathematics does and does not study.

My proposed sentence on the other hand is not a definition -- is says that mathematics is a "large and varied field of study" which is descriptive, not definitional, because it says nothing about what it is studying or not studying: is it numbers? Frogs? Zeus? From there it just talks about the historical context to give direction, and relies on the rest of the introduction to give enough examples of mathematical areas of interest to hopefully provide a fuller context.

Lauciusa (talk) 16:47, 27 March 2025 (UTC)

The current first sentence is descriptive. That's the point. Tito Omburo (talk) 17:25, 27 March 2025 (UTC)

The reason why I say prescriptive rather than descriptive is that the existing opening rules out some things. If a subject is not developed or proved for the empirical sciences nor for mathematics itself, then it is not part of mathematics. So methods and calculations designed for record keeping or actuarial uses are not mathematics. This is simply not true; arguably these are the origin of the field and the way that it is taught in primary and secondary school to this day.

On the other hand, my introduction draws a very big circle of things that it could study, and gives two members of that set; members that are, in particular, the origins of the entire field of study.

Lauciusa (talk) 18:34, 28 March 2025 (UTC)

Lauciusa, I'm really having trouble following you here. It seems to me your proposal says more about the objects of study, at least originally ("numbers and shapes"), whereas the current first sentence says absolutely nothing about the objects of study, and instead describes the methodology. That was what I actually liked better about your proposal. (But as Remsense correctly points out, this is less of an issue with the current lead if you also consider the second sentence.)

In any case, neither version seems to be a definition. --Trovatore (talk) 06:01, 28 March 2025 (UTC)

See my reply to Tito Omburo just above this. My problem with the current definition is that it is inaccurate. My alternative is admittedly overly broad, but is not incorrect, and at least gives a motivation for the field in historical terms.

Lauciusa (talk) 18:42, 28 March 2025 (UTC)

How about:

Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of analysis and manipulation of numbers, shapes, and other kinds of appropriate objects.

Paul August ☎ 13:43, 28 March 2025 (UTC)

I guess I don't hate it, but would prefer "quantities" to "numbers". Tito Omburo (talk) 15:36, 28 March 2025 (UTC)

What "appropriate object" is intended to mean, and for which audience this phrase is meaningful?

For avoiding apparent circularity, I suggest to replace "mathematics itself" with "previously established parts of mathematics". This would give the first sentence Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and previously established parts of mathematics. D.Lazard (talk) 16:45, 28 March 2025 (UTC)

Thinking again, I do not like either "field of study that discovers and organizes" (almost each term is more or less controversial). So, I suggest: Mathematics deals with abstract methods, theories and theorems that are developed and proved for the needs of empirical sciences and previously established parts of mathematics. D.Lazard (talk) 16:54, 28 March 2025 (UTC)

I like the "previously established"; it's overly formal but does address the recursiveness. I also like the "deals with" which is less limiting than the current language. If we adopted this phrasing it would be a strict improvement.

I feel like the connection to the empirical sciences is a relatively recent development in the history of mathematics and does not deserve to be in the opening. The answer to what particular aspects of the real world led to the development of mathematics is unaddressed while I feel like my opening does at least indicate some intuition about where mathematics is used.

Lauciusa (talk) 18:46, 28 March 2025 (UTC)

The earliest mathematics was used in commerce and architecture by the Babylonian civilization, primarily for reckoning lengths and weights. That is, it was used as a tool to describe the world. Tito Omburo (talk) 19:46, 28 March 2025 (UTC)

I am not a huge fan of this phrasing because I feel it is too limiting, but I love the idea of using "quantities" instead of "numbers". Lauciusa (talk) 18:16, 28 March 2025 (UTC)

There is only so much one can do in 80 words: I would prioritize providing meaningful description over that which endeavors to completely circumscribe at the expense of being overly vague. Really, the description is the entire article. We should favor providing as much "survey coverage" as quickly as possible in the lead. Remsense ‥ 论 18:27, 28 March 2025 (UTC)

Introduction counter-proposal

We argue incessantly (actually: periodically with high frequency) about the opening sentence. It's so over-constrained that no solution is ever entirely satisfying. The most recent proposal by Paul August is admirable but typical: It attempts to say everything important, and thus becomes a complicated nest of multiple dependent clauses. I believe that such sentences are off-putting to many native speakers and incomprehensible to many non-native speakers. The opening sentence often resembles a definition, which invites further complaints, even though we disavow it as a definition.

In the interest of breaking this cycle, here is my radical proposal: We make the opening sentence so short, that it can't possibly be taken as a definition or summary of math. Rather, it is obviously just the start of a paragraph. The paragraph has enough room, to hold everything that we wanted to put in the opening sentence.

So, for those of you willing to consider this alternative approach, here is a proposed opening sentence:

Mathematics is a field of study.

Mgnbar (talk) 21:14, 28 March 2025 (UTC)

Thanks, but I hate it. ;-) Tito Omburo (talk) 21:22, 28 March 2025 (UTC)

I totally disagree, since mathematics is also a field of knowledge, and knowledge is not included in study, even if study is required to obtain knowledge. This is one of the things I had in mind by saying that almost all words that I suggest to replace with "deals with" are controversial. D.Lazard (talk) 21:40, 28 March 2025 (UTC)

I just took "field of study" from the current version at Mathematics. Other, similarly short formulations fit my proposal equally well. Mgnbar (talk) 22:15, 28 March 2025 (UTC)